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An Inertial Three-term Derivative-free Projection Algorithm for Nonlinear Equations without Pseudo-monotonicity

Author

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  • Xiaoyu Wu

    (Jiangsu Center for Applied Mathematics, China University of Mining and Technology)

  • Hu Shao

    (Jiangsu Center for Applied Mathematics, China University of Mining and Technology)

  • Pengjie Liu

    (Jiangsu Center for Applied Mathematics, China University of Mining and Technology)

  • Feng Shao

    (Jiangsu Center for Applied Mathematics, China University of Mining and Technology)

Abstract

In this paper, we focus on developing a general form of inertial iterative method for the system of unconstrained nonlinear equations, which has extensive and practical applications. Combining the inertial step and projection technique, a family of three-term conjugate gradient projection method is proposed for finding the approximate solution of nonlinear equations. The search direction is modified based on the scaled memoryless BFGS formula, satisfying the sufficient descent property. Our methods are suitable for large-scale equations since they are low storage memory and derivative-free. Moreover, we analyze the global convergence of the proposed method without the monotonicity or pseudo-monotonicity as well as the Lipschitz continuity hypothesis of the system of nonlinear equations. Additionally, the numerical experiments on nonlinear equations and applications on compressed sensing problems are conducted to verify the effectiveness of our methods.

Suggested Citation

  • Xiaoyu Wu & Hu Shao & Pengjie Liu & Feng Shao, 2025. "An Inertial Three-term Derivative-free Projection Algorithm for Nonlinear Equations without Pseudo-monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 206(2), pages 1-30, August.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:2:d:10.1007_s10957-025-02711-7
    DOI: 10.1007/s10957-025-02711-7
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    References listed on IDEAS

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    1. Gonglin Yuan & Zehong Meng & Yong Li, 2016. "A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 129-152, January.
    2. Predrag S. Stanimirović & Branislav Ivanov & Snežana Djordjević & Ivona Brajević, 2018. "New Hybrid Conjugate Gradient and Broyden–Fletcher–Goldfarb–Shanno Conjugate Gradient Methods," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 860-884, September.
    3. David F. Shanno, 1978. "Conjugate Gradient Methods with Inexact Searches," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 244-256, August.
    4. Jamilu Abubakar & Poom Kumam & Habib ur Rehman & Abdulkarim Hassan Ibrahim, 2020. "Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator," Mathematics, MDPI, vol. 8(4), pages 1-25, April.
    5. XiaoLiang Dong & Deren Han & Zhifeng Dai & Lixiang Li & Jianguang Zhu, 2018. "An Accelerated Three-Term Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 944-961, December.
    6. Abubakar, Auwal Bala & Kumam, Poom & Ibrahim, Abdulkarim Hassan & Chaipunya, Parin & Rano, Sadiya Ali, 2022. "New hybrid three-term spectral-conjugate gradient method for finding solutions of nonlinear monotone operator equations with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 201(C), pages 670-683.
    7. Xiaoyu Wu & Hu Shao & Pengjie Liu & Yue Zhuo, 2023. "An Inertial Spectral CG Projection Method Based on the Memoryless BFGS Update," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1130-1155, September.
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